Lcomoment.coefficients {lmomco} | R Documentation

## L-comoment Coefficient Matrix

### Description

Compute the L-comoment coefficients from an L-comoment matrix of order k >= 2 and the k = 2 (2nd order) L-comoment matrix. However, if the first argument is 1st-order then the coefficients of L-covariation are computed. The function requires that each matrix has already computed by the function `Lcomoment.matrix`.

### Usage

    Lcomoment.coefficients(Lk,L2)

### Arguments

`Lk` | A k >= 2 L-comoment matrix from `Lcomoment.matrix`.
`L2` | A k = 2 L-comoment matrix from `Lcomoment.matrix(Dataframe,k=2)`.

### Details

The coefficient of L-variation is computed by `Lcomoment.coefficients(L1,L2)` where `L1` is a 1st-order L-moment matrix and `L2` is a k = 2 L-comoment matrix. Symbolically, the coefficient of L-covariation is

hat{tau}_{[12]} = frac{hat{λ}_{2[12]}} {hat{λ}_{1[12]}} mbox{.}

The higher L-comoment coefficients (L-coskew, L-cokurtosis, ...) are computed by the function `Lcomoment.coefficients(L3,L2)` (k=3), `Lcomoment.coefficients(L4,L2)` (k=4), and so on. Symbolically, the higher L-comoment coefficients are

hat{tau}_{k[12]} = frac{hat{λ}_{k[12]}} {hat{λ}_{2[12]}} mbox{, for } k >= 3 mbox{.}

Finally, the usual univariate L-moment ratios as seen from `lmom.ub` or `lmoms` are along the diagonal. The `Lcomoment.coefficients` function does not make use of `lmom.ub` or `lmoms`.

### Value

An R `list` is returned.

`type` | The type of L-comoment representation in the matrix: “Lcomoment.coefficients”.
`order` | The order of the coefficients. `order=2` L-covariation, `order=3` L-coskew, ...
`matrix` | A k >= 2 L-comoment coefficient matrix.

### Note

The function begins with a capital letter. This is intentionally done so that lower case namespace is preserved. L-comoments are new in the literature and experimental in this package. By using a capital letter now, then `lcomoment.coefficients` remains an available name in future releases.

### Author(s)

W.H. Asquith

### Source

Serfling and Xiao (2006).

### References

Serfling, R., and Xiao, P., 2006, Multivariate L-moments, preprint.

### See Also

`lmom.ub`, `Lcomoment.matrix`, `Lcomoment.coefficients`

### Examples

    D      <- data.frame(X1=rnorm(30),X2=rnorm(30),X3=rnorm(30))
    L1     <- Lcomoment.matrix(D,k=1)
    L2     <- Lcomoment.matrix(D,k=2)
    L3     <- Lcomoment.matrix(D,k=3)
    LkLCV  <- Lcomoment.coefficients(L1,L2)
    LkTAU3 <- Lcomoment.coefficients(L3,L2)

[Package lmomco version 0.92 Index]